Problem 65 - College Algebra and Calculus: An Applied Approach

Question

The revenue \(R\) and cost \(C\) for a product are given by \(R=x(50-0.0002 x)\) and \(C=12 x+150,000\), where \(R\) and \(C\) are measured in dollars and \(x\) represents the number of units sold (see figure). (a) How many units must be sold to obtain a profit of at least \(\$ 1,650,000 ?\) (b) The demand equation for the product is \(p=50-0.0002 x\) where \(p\) is the price per unit. What prices will produce a profit of at least \(\$ 1,650,000 ?\) (c) As the number of units increases, the revenue eventually decreases. After this point, at what number of units is the revenue approximately equal to the cost? How should this affect the company's decision about the level of production?

Step-by-Step Solution

Answer

The number of units that must be sold for a profit of $1650000, the price per unit that produces such profit and the number of units at which revenue approximately equals cost, should drive the decision on the level of production.

Step 1: Calculate the number of units to achieve a specified profit

The profit \( P \) is given by the formula \( P = R - C \). We can set up the equation \( P = R-C = x(50-0.0002x) - (12x + 150000) \geq 1650000 \) and solve it for \( x \).

Step 2: Determine the price per unit for a given profit

The equation \( p = 50 - 0.0002x \) describes the price per unit. From step 1, we will have the value of \( x \) that gives a profit of at least $1650000. Substituting this value in the price equation will give us the price per unit.

Step 3: Find the number of units at which revenue equals to cost

For this, we set \( R = C \) and solve for \( x \). That is, we solve the equation \( x(50-0.0002x) = 12x + 150000 \). The roots of this equation will give the number of units at which revenue is approximately equal to the cost.

Step 4: Analyzing the implication of the solution

Based on the solutions obtained in the previous steps, a brief analysis can be made on how the solutions should affect the company's decision about the level of production.

Key Concepts

Revenue Function

In the world of business, the revenue function is crucial as it represents the income generated from selling goods or services. Here, our revenue function is given by the formula \( R = x(50 - 0.0002x) \), where \( R \) stands for revenue and \( x \) signifies the number of units sold. This equation helps determine how much money a business makes based on the number of items sold.

• The term \( 50 - 0.0002x \): This is the pricing formula that adjusts the price per unit based on the quantity sold. As more units are sold, the price slightly decreases, reflecting a common market situation where prices drop due to increased supply.
• The variable \( x \): Represents the quantity of goods sold and directly influences the revenue.
• Maximizing Revenue: Finding the optimal number of units **x** that keeps increasing revenue, requires examining the relationship between revenue and units sold.

Balancing the made revenue with strategic pricing and quantity can help a business maintain profitability and competitiveness in the market.

Cost Function

Understanding the cost function is essential for businesses to manage expenses and predict profitability. In our scenario, the cost function is formulated as \( C = 12x + 150,000 \). This formula shows the costs involved in producing and selling a certain number of goods \( x \).

• Fixed cost: The second part of the equation, \( 150,000 \), represents the fixed costs, which are expenses that do not change regardless of the production level. This could include rent, salaries, and other overheads.
• Variable cost: The \( 12x \) term signifies the variable cost, which changes with the level of production. Each unit produced increases the total cost by $12, covering more direct expenses like materials or labor.

Knowing these costs helps businesses plan how to offset expenses with adequate sales, ensuring sustainability and profitability. Market conditions must be evaluated to make strategic production decisions.

Demand Equation

The demand equation describes how the price of a product changes with the number of units sold and is defined here as \( p = 50 - 0.0002x \). Understanding the demand relationship is pivotal for setting appropriate price points.

• Price per unit: The equation indicates that as more units are sold, the price drops. This can be viewed as a pricing strategy to boost sales volume.
• Market Behavior: This is a classic supply and demand model, where increased supply (more units sold) generally leads to a lower price to entice more buyers.

By using this equation, businesses determine optimal pricing that balances profit margins with attractive market rates. Adjusting prices will depend on sales data, costs, and desired revenue goals.

Break-even Point

The break-even point is a critical concept that identifies when a business's revenues will fully cover its costs, leading to neither profit nor loss. It's calculated where the revenue equals the cost, thus \( R = C \). For our case, this involves solving \( x(50 - 0.0002x) = 12x + 150,000 \).

• Calculation: To find this point, we rearrange the function to solve for \( x \), which will tell us the number of units needed to break even.
• Importance: It helps businesses set sales targets and review cost management strategies. Knowing when costs match revenue is vital for financial planning and risk assessment.
• Decision Making: Beyond the break-even point, each unit sold adds profit, underscoring the importance of surpassing this level in business.

Reaching and surpassing this point guarantees that a business can cover its expenses and begin generating profit.